Our two main theorems, the first one stating the equivalence of \(M_1\Rightarrow _{\mathsf {lex}} \mathsf {Mal}\)  and \(M_1\Rightarrow _{\mathsf {alg}} \mathsf {Mal}\)  for a simple extended matrix \(M_1\)  and the second one stating the equivalence of \(M_1\Rightarrow _{\mathsf {reg}} \mathsf {Mal}\)  and \(M_1\Rightarrow _{\mathsf {alg}} \mathsf {Mal}\)  for a (general) extended matrix \(M_1\)  are quite surprising and particular to the Mal'tsev case. Indeed, as it is the general philosophy of the papers [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, to prove the validity of many statements about exactness properties, one is often required to produce a proof in the essentially algebraic context (and not just in the varietal context as it is the case in the present situation). Actually, we prove these two theorems not only for the Mal'tsev matrix \(\mathsf {Mal}\) , but for the matrix \(\mathsf {Cube}_n\)  for each \(n\geqslant 2\) , describing the Mal'tsev condition of having an \(n\) -cube term [8]}. The Mal'tsev case is then recovered in the case \(n=2\) .
